In medical follow-up and life testing, one gets to observe randomly censored life times. For example, the subject may leave the study or survive till the completion of the study. It is proposed to develop the tests on the basis of randomly censored data of the hypothesis H: the life distribution is exponential against the alternatives 1) H1: the life distribution is New Better Than Used in Expectation, not an exponential, (2) H2: the life distribution is New Better than Used, not an exponential and (3) H3: the life distribution is IFR. We also propose to make the Bahadur Efficiency comparisons of various tests. Our frame work will be nonparametric. Our tests will be based on the Bayes estimators of the Susarla-Van Ryzin type and methodology of proving various results will include the weak convergence techniques and the large deviation results and some Monte Carlo studies. These testing problems should be interesting from the management point of view. For example, if one would know that the life distribution is NBU, then one may prefer an age or block replacement policy to an ordinary replacement policy. This is so because a life distribution is NBU if and only if the number of replacements under an ordinary replacement plan is stochastically larger than that under the age or block replacement plan. The Bahadur efficiency compares sample sizes required to obtain arbitrarily small significance level at the fixed alternatives and power. A procedure with higher exact Bahadur slope than the other procedure would yield arbitrarily small significance level more cheaply than other and such a procedure should be of value in the above situations.